Timeline for Does unique continuation also hold for derivatives of solutions?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 5, 2021 at 18:50 | vote | accept | Leo Moos | ||
| Jun 5, 2021 at 18:48 | answer | added | Willie Wong | timeline score: 2 | |
| Jun 5, 2021 at 18:33 | comment | added | Leo Moos | @WillieWong Shoot, maybe my question was unclear after all; sorry about that. I really meant the partial derivatives, not the whole gradient. Put differently, the question would be something like 'If the partial derivative $D_k u$ has infinite vanishing order at the origin, then $D_k u \equiv 0$'. | |
| Jun 5, 2021 at 18:19 | comment | added | Willie Wong | In fact, if $c$ is constant, the same argument works, as you just add $c \vec{v}$ to the equation above. | |
| Jun 5, 2021 at 18:07 | comment | added | Willie Wong | If you let $\vec{v} = \nabla u$ be the vector valued functions, it satisfies an equation of the form $(a \cdot \nabla^2) \vec{v} + (\nabla a + b) \cdot \nabla \vec{v} + \nabla b \cdot \vec{v} = 0$. As the principal part is diagonal, you can apply the scalar Carleman estimates (e.g. jstor.org/stable/2374732) and conclude strong unique continuation. | |
| Jun 5, 2021 at 6:54 | comment | added | Leo Moos | @WillieWong That's good to keep in mind, thanks - I've seen similar examples in a couple of papers. Regarding the second part, if e.g. $c \equiv 0$ would there be an affirmative answer to the question? | |
| Jun 4, 2021 at 20:09 | comment | added | Willie Wong | If you don't assume $u(0) = 0$, then you have easy counterexamples: take any positive smooth function that is constant near the origin. Define $c = \frac{1}{u}(a^{ij}\partial^2_{ij} u + b^i \partial_i u)$. This example can be ruled out with with additional hypotheses on $c$. | |
| Jun 4, 2021 at 18:20 | comment | added | Leo Moos | @WillieWong I've reworded the question slightly - is it clearer now? I think your second comment is precisely what I meant to ask. Could you share the example you have in mind? | |
| Jun 4, 2021 at 18:18 | history | edited | Leo Moos | CC BY-SA 4.0 |
clarified question
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| Jun 4, 2021 at 16:26 | comment | added | Willie Wong | In particular, if your question is: if $u$ solves a second order linear elliptic PDE as above, and you know that $Du$ vanishes to infinite order at the origin, is $u$ constant? Then the answer is "not in general". | |
| Jun 4, 2021 at 16:22 | comment | added | Willie Wong | Can you clarify your question? Does what hold under what hypotheses? It is not clear from your question which hypotheses you want to share between the known result and that which you want to ask about. | |
| Jun 4, 2021 at 15:46 | history | asked | Leo Moos | CC BY-SA 4.0 |